Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6354266 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C22×#121 |
| commutator, Frattini, cosocle | C42 | C25 |
| prime radical | #3444 | C22 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
1565 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2,7,11)(3,12,14,15)(4,9,8,16)(5,10,6,13)(17,18)(19,25)(20,28)(21,23)(22,26)(24,30)(27,32)(29,31);(1,3,6,9,7,14,5,16)(2,8,10,12,11,4,13,15);(1,4)(2,9)(3,13)(5,12)(6,15)(7,8)(10,14)(11,16)(17,19,21,27)(18,22,23,31)(20,29,30,26)(24,32,28,25);(1,5,7,6)(2,10,11,13)(3,9,14,16)(4,12,8,15)(18,23)(22,31)(24,28)(25,32);(3,14)(4,8)(9,16)(12,15)Polycyclic presentation:
2^9|a^2=h,b^2=f,c^2=i,d^2=h,f^2=h,g^2=i,b^a=bf,c^a=cg,d^a=di,f^a=fh,g^a=gi,d^b=dh,e^b=eh,e^c=eh,g^c=gi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72306718, #72306720, #72306722, … |
| maximal normal subgroup of | #72306718, #72306720, #72306722, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6261469);